Method of out-of-band correction for multispectral remote sensing

ABSTRACT

A method of image processing. A band-averaged spectral radiance is measured using at least one optical filter upon scanning a plurality of original radiances. The measured band-averaged spectral radiance includes a measured in-band-averaged spectral radiance and a measured band-gap-averaged spectral radiance. A multispectral radiance vector is generated from the measured band-averaged spectral radiance. The multispectral radiance vector and an out-of-band correction transform matrix corresponding to the at least one optical filter are matrix-multiplied to generate a band-averaged spectral radiances image vector representing a plurality of recovered band-averaged spectral radiances. The plurality of recovered band-averaged spectral radiances is analyzed for a presence of a target.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application claims priority to U.S. Provisional PatentApplication Ser. No. 61/674,954, which was filed on 24 Jul. 2012.Additionally, the present application is a continuation-in-partapplication of U.S. patent application Ser. No. 13/862,539, which wasfiled 15 Apr. 2013.

FIELD OF THE INVENTION

The invention relates generally to a method of image processing, andmore particularly to a method of multispectral decomposition for theremoval of out-of-hand effects.

BACKGROUND OF THE INVENTION

Multispectral remote sensing images are acquired from aircraft andsatellites. To quantify ground surface characteristics, the measuredspectral radiances must be converted into target reflectance. In theseapplications, accurate and consistent sensor calibration is essential.Out-Of-Band (“OOB”) response is defined as the ratio of integratedresponse outside the one percent of peak response points of a spectralband to the integrated response inside the one percent points. Severalmultispectral radiometric instruments are known to exhibit significantradiance contribution from OOB spectral response.

The typical scale of OOB spectral response is in the range of severalpercent, which can, for example, result in chlorophyll retrievals thatare biased high for clear water by OOB response to short wavelengths. Amethodology to dealing with the OOB response had been suggested andadopted for SeaWiFS calibration. These calibration methods adjust themeasured radiances to correct for OOB response for ease of comparison toin situ measured multispectral radiances. The SeaWiFS correction schemehas been successfully applied to data products retrieved over Case 1ocean waters. “Case 1” ocean waters are those for which the inherentoptical properties are determined primarily by phytoplankton andco-varying chromophoric dissolved organic matter (“CDOM”) and detritus.However, the correction scheme is inherently not useable for SeaWiFSdata product corrections over Case 2 turbid waters or over land.

The first VIIRS instrument, Flight Unit 1 (“FU1”), now flying on the NPPsatellite platform, has known performance issues. Seven channels locatedbetween 0.4 and 0.9 μm in the VisNIR focal plane have problems relatedto OOB responses, i.e., small amounts of radiance far away from thecenter of a given channel that pass through the filter and reach thedetector. The newly launched VIIRS instrument requires developing highlyaccurate operational calibration procedures and algorithms to processVIIRS data.

BRIEF SUMMARY OF THE INVENTION

An embodiment of the invention includes a method of recovering thein-band multispectral radiances and addressing the issues of the OOBresponse. For a particular multispectral channel, other channels providemeasurements of spectral regions that contribute OOB radiance. Thiscrosstalk between multispectral channels provides a possibility forcorrection. The new approach is based on the decomposition principle torecover the average narrowband signals from uncorrected signals usingfilter transmittance functions instead of the calibration methods.

In this alternative embodiment of the invention, using thelaboratory-measured filter transmittance functions for all multibandchannels, an out-of-band correction transform (“OBCT”) matrix forrecovering in-band spectral radiances is derived. For an N-channelmultispectral sensor, OOB effects are corrected b applying an N×N OBCTmatrix to the measured signals.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart of an illustrative method embodiment of theinstant invention.

FIGS. 2A-2G are graphs of VIIRS version 3 M1-M7 filter transmittancecurves, respectively, normalized at the peaks.

FIGS. 3A and 3B are graphs showing an in-band wavelength range and bandminimum and maximum wavelength positions for normalized-to-peak VIIRS M4and M6 filter transmittance curves, respectively.

DETAILED DESCRIPTION OF THE INVENTION

An embodiment of the invention includes a method described as follows,with reference to FIG. 1. A band-averaged spectral radiance is measuredusing at least one optical filter upon scanning a plurality of originalradiances, as in Step S100. The measured band-averaged spectral radianceincludes a measured in-band-averaged spectral radiance and a measuredband-gap-averaged spectral radiance. A multispectral radiance vector isgenerated from the measured band-averaged spectral radiance, as in StepS110. The multispectral radiance vector and an out-of-band correctiontransform matrix corresponding to the at least one optical filter arematrix-multiplied to generate a band-averaged spectral radiances imagevector representing a plurality of recovered hand-averaged spectralradiances, as in Step S120. The plurality of recovered band-averagedspectral radiances is analyzed for a presence of a target, as h StepS130. Illustrative targets include any matter, such as, ocean featuresor land features).

Optionally, the at least one optical filter resides on one an aircraftand a satellite.

Optionally, the hand-averaged spectral radiance includes a VIIRShand-averaged spectral radiance.

Another embodiment of the invention includes a method described asfollows. A band-averaged spectral radiance is measured using at leastone optical filter, upon scanning a plurality of original radiances, themeasured total band-averaged spectral radiance, ŝ_(k)=ŝ_(k)(i,j)=∫_(λe|Δλj)h_(k)(λ)s(λ)dλ+∫_(λ∉{Δλj})h_(k)(λ)s(λ)dλ, where i and j arepixel indexes, comprising a measured in-band-averaged spectral radianceand a measured band-gap-averaged spectral radiance, the measured totalband-averaged spectral radiance comprising a plurality of k wavelengthsub-ranges, the plurality of wavelength sub-ranges comprising aplurality of k in-band sub-ranges and a plurality a plurality of kband-gap sub-ranges, wherein a measured kth in-band band-averagedspectral radiance is represented as

${{\int_{\lambda \in {\{{\Delta\;\lambda_{l}}\}}}{{h_{k}(\lambda)}{s(\lambda)}\ {\mathbb{d}\lambda}}} \approx {\sum\limits_{l = 1}^{N}{{\overset{\_}{h}}_{k\; l}^{({i\; n})}\Delta\;\lambda_{l}{\overset{\_}{s}}_{l}}}},$

where N is a number of a plurality of bands,

h _(kl) ^((in)) is an average of a plurality of filter responsefunctions,

Δλ₁ is a width of partitioned sub-band, and

s ₁ is a recovered Ith band-average spectral radiance that is an averageof all measured signals within the sub-band Δλ₁, and free of anout-of-band effect,

wherein a measured k-th band-gap band-averaged spectral radiance isrepresented as

$\mspace{79mu}{{{\int_{\lambda \notin {\{{\Delta\;\lambda_{l}}\}}}{{h_{k}(\lambda)}{\hat{s}(\lambda)}\ {\mathbb{d}\lambda}}} \approx {\sum\limits_{l = 1}^{N}{b_{k\; l}{\hat{s}}_{l}}}},\mspace{79mu}{where}}$$b_{kl} = \left\{ {{{\begin{matrix}{{\left( {\lambda_{\min}^{(1)} - \lambda_{\min}} \right){\overset{\_}{h}}_{k,0}^{({out})}} + {\frac{\lambda_{\min}^{(2)} - \lambda_{\max}^{(1)}}{\lambda_{2} - \lambda_{1}}\left( {{\lambda_{2}{\overset{\_}{h}}_{k,1}^{({out})}} - {\overset{\_}{h\;\lambda}}_{k,1}^{({out})}} \right)}} & \left( {l = 1} \right) \\\begin{matrix}{{\frac{\lambda_{\min}^{(l)} - \lambda_{\max}^{({l - 1})}}{\lambda_{l} - \lambda_{l - 1}}\left( {{\overset{\_}{h\;\lambda}}_{k,{l - 1}}^{({out})} - {\lambda_{l - 1}{\overset{\_}{h}}_{k,{l - 1}}^{({out})}}} \right)} +} \\{\frac{\lambda_{\min}^{({l + 1})} - \lambda_{\max}^{(l)}}{\lambda_{l + 1} - \lambda_{l}}\left( {{\lambda_{l + 1}{\overset{\_}{h}}_{k\; l}^{({out})}} - {\overset{\_}{h\;\lambda}}_{k\; l}^{({out})}} \right)}\end{matrix} & {\left( {1 < l < N} \right),} \\\begin{matrix}{{\frac{\lambda_{\min}^{(N)} - \lambda_{\max}^{({N - 1})}}{\lambda_{N} - \lambda_{N - 1}}\left( {{\overset{\_}{h\;\lambda}}_{k,{N - 1}}^{({out})} - {\lambda_{N - 1}{\overset{\_}{h}}_{k,{N - 1}}^{({out})}}} \right)} +} \\{\left( {\lambda_{\max} - \lambda_{\max}^{(N)}} \right){\overset{\_}{h}}_{k\; N}^{({out})}}\end{matrix} & \left( {l = N} \right)\end{matrix}\mspace{20mu}{\overset{\_}{h\;\lambda}}_{kl}^{({out})}} = \frac{\int_{\lambda_{(\max)}^{(l)}}^{\lambda_{\min}^{({l + 1})}}{{h_{k}(\lambda)}\lambda\ {\mathbb{d}\lambda}}}{\lambda_{\min}^{({l + 1})} - \lambda_{\max}^{(l)}}},\mspace{20mu}{{{and}\mspace{20mu}{\overset{\_}{h}}_{k\; l}^{({out})}} = {\frac{\int_{\lambda_{(\max)}^{(l)}}^{\lambda_{\min}^{({l + 1})}}{{h_{k}(\lambda)}{\mathbb{d}\lambda}}}{\lambda_{\min}^{({l + 1})} - \lambda_{\max}^{(l)}}.}}} \right.$A multispectral radiance vector is generated from the measuredband-averaged spectral radiance. The multispectral radiance vector andan out-of-hand correction transform matrix corresponding to the at leastone optical filter are matrix-multiplied to generate a band-averagedspectral radiances image vector representing a plurality of recovered baid-averaged spectral radiances. The plurality of recovered band-averagedspectral radiances are outputted, thereby generating a plurality ofrecovered radiances free of out-of-band effects.

Optionally, wherein each in-hand sub-range of the plurality of k in-bandsub-ranges comprises an in-band width and each band-gap sub-ranges ofthe plurality of k in-band sub-ranges includes an hand-gap width,wherein the at least one optical filter comprises at least one filtertransmittance function, wherein the plurality of sub-ranges comprises atleast one in-band partition parameter, and wherein the multispectraldecomposition transform matrix is a function of at least one of the atleast one filter transmittance function the at least one partitionparameter, and a position of the at least one optical filter.

Optionally, the at least one optical filter comprises a number ofmulti-bands, the number of multi-bands being equal to a number of theplurality of sub-ranges.

Optionally, the recovered and measured band-averaged spectral radiancevector are represented as

${\overset{\_}{s} = \begin{pmatrix}{\overset{\_}{s}}_{1} \\{\overset{\_}{s}}_{2} \\\ldots \\{\overset{\_}{s}}_{N}\end{pmatrix}},{{{and}\mspace{11mu}\hat{s}} = \begin{pmatrix}{\hat{s}}_{1} \\{\hat{s}}_{2} \\\ldots \\{\hat{s}}_{N}\end{pmatrix}}$

wherein each components s _(n) and ŝ_(n) of the recovered and measuredband-averaged spectral radiance vectors are single band images.

Optionally, the band-averaged spectral radiance image vector isrepresented s=Tŝ, wherein the out-of-band correction transform matrix Tis defined byT=A ⁻¹(I−B),

wherein matrix A=(h _(kl) ^((in))Δλ₁) and matrix) are N×N,

wherein matrix I is a N×N identity matrix,

wherein ŝ is the measured band-averaged spectral radiance vector.

Optionally, wherein the band-averaged spectral radiance comprises aVIIRS band-averaged spectral radiance.

An embodiment of the invention is described as follows.

Put-Of-Band Correction Transform

Multiband Radiometric Instrument

VIIRS is a typical multispectral remote sensing instrument. Throughvarious laboratory tests of the first VIIRS instrument, it has beenfound that the seven channels located between 0.4 and 0.9 μm (M1-M7) inthe VisNIR focal plane have problems with OOB responses. A set of VIIRS(Version 3) M1-M7 filter transmittance curves (normalized at the peak ofthe filter transmission) is shown in FIGS. 3A-3G. The VIIRS filter dataare available from the public domain websitehap://www.star.nesdis.noaa.giov/jpss/index.php. The M1 and M4 filtercurves peak at wavelengths below 0.6 μm on the left side of the plot,and both have significant transmittances in the far distant. “wing”region above 0.6 μm.

The VIIRS VisNIR channel names, positions, and widths are listed inTable 1 below. Many VIIRS channels (designated as M1 to M7 in Table 1)have heritages to the MODIS instrument but with minor differences incenter positions and widths.

TABLE 1 VIIRS VisNIR CHANNEL NAMES, POSITIONS, AND FULL WIDTHS AT HALFMAXIMUM (FWHMS). VIIRS Channel λ (μm) FWHM (μm) M1 0.412 0.020 M2 0.4450.018 M3 0.488 0.020 M4 0.555 0.020 M5 0.672 0.020 M6 0.746 0.015 M70.865 0.039

The causes for the OOB response with VIIRS M1-M7 channels are now fullyunderstood. The main cause for the OOB response is associated withhigh-angle scattering in the integrated filter assembly that overliesthe VisNIR focal plane array. The scattering mechanism causes the OOBeffects for a given channel to come from a broad spectral range, insteadof a few narrow spectral intervals.

A. Linear Optical System

In general, a multi-spectral instrument such as VIIRS is considered tobe a system that accepts an input, and produces an output. Such a systemis linear, because the measured optical single band signal[ŝ_(k)=ŝ_(k)(i,j), where i and j are pixel indexes] from a sensor withthe k^(th) band filter on a pixel can be expressed byŝ _(k)=∫_(λ) _(xixi) ^(λ) ^(xixi) h _(k)(λ)s(λ)dλ,   (1)where ŝ_(k) and s(λ) are a measured band-averaged spectral radiances(with OOB effects) and original radiances, respectively, and h_(k)(λ)are the spectral response functions of the optical system (filters) withthe wavelength λ ∈ (λ_(min), λ_(max)) as a variable, where (λ_(min),λ_(max)) is, for VIIRS, the entire VisNIR spectral range. The spectralresponse functions h_(k)(λ) are normalize(between the full rangewavelength λ_(min) and λ_(max) as follows∫_(λ) _(xix) ^(λ) ^(xxx) h _(k)(λ)dλ=1.   (2)The above superposition integral expresses a relationship betweenoriginal and measured signals with the optical filters.

The full range integral in equation (1) between the cut-off wavelengthsλ_(min) and λ_(max) can be partitioned by two parts in which thewavelength ranges cover in-band (narrow bandwidths with nominal bandcenters λ_(k) in Table 1) regions and band-gap regions between in-bandregions, respectively. If the in-band wavelength width Δλ₁=ζ_(max)⁽¹⁾−λ_(min) ⁽¹⁾ is defined by a spectral response function as shown inFIGS. 3A and 3B, the integral in (1) becomesŝ _(k)=∫_(λ∉{Δλ1) }h _(k)(λ)s(λ)dλ+∫ _(λ∉{Δλ) }h _(k)(λ)s(λ)dλ,   (3)where {Δλ₁}=(Δλ₁, Δλ₂, . . . , Δλ_(N)) denotes the in-band range of allchannels, and N is the number of filters. The bandwidths Δλ₁ are not thesame bandwidths defined in Table 1. The bandwidths Δλ₁ are usuallyselected to be slightly greater than bandwidths for which the responseis inside the 1% of peak response points, and depend on characteristicsof the response functions of the filters.In-Band Partitions

If the number of filters is equal to N, then the in-band integral inequation (3) is given by

$\begin{matrix}{{\int_{\lambda \in {\{{\Delta\;\lambda_{l}}\}}}{{h_{k}(\lambda)}{s(\lambda)}\ {\mathbb{d}\lambda}}} = {\sum\limits_{l = 1}^{N}{\int_{\lambda_{\min}^{(l)}}^{\lambda_{\max}^{(l)}}{{h_{k}(\lambda)}{s(\lambda)}\ {{\mathbb{d}\lambda}.}}}}} & (4)\end{matrix}$Using an average value of the response function between λ_(min) ⁽¹⁾ andλ_(max) ⁽¹⁾ to replace the response function h_(k)(λ) in the integral,we have

$\begin{matrix}{{{{\int_{\lambda_{\min}^{(l)}}^{\lambda_{\max}^{(l)}}{{h_{k}(\lambda)}{s(\lambda)}\ {\mathbb{d}\lambda}}} \approx {{\overset{\_}{h}}_{k\; l}^{({i\; n})}{\int_{\lambda_{\min}^{(l)}}^{\lambda_{\max}^{(l)}}{{s(\lambda)}{\mathbb{d}\lambda}}}}} = {{\overset{\_}{h}}_{k\; l}^{({i\; n})}\Delta\;\lambda_{l}{\overset{\_}{s}}_{l}}},} & (5)\end{matrix}$where Δλ₁=λ_(max) ⁽¹⁾−λ_(min) ⁽¹⁾ and the average of the responsefunctions is given by

${{\overset{\_}{h}}_{k\; l}^{({i\; n})} = {\frac{1}{\Delta\;\lambda_{l}}{\int_{\lambda_{\min}^{(l)}}^{\lambda_{\max}^{(l)}}{{h_{k}(\lambda)}{\mathbb{d}\lambda}}}}},$and the in-band signal [s ₁=s ₁(i, j) where i and j are pixel indexes]for a particular band that is an average of all signals within thesub-band Δλ₁ defined by

$\begin{matrix}{{\overset{\_}{s}}_{l} = {\frac{1}{\Delta\;\lambda_{l}}{\int_{\lambda_{\min}^{(l)}}^{\lambda_{\max}^{(l)}}{{s(\lambda)}{{\mathbb{d}\lambda}.}}}}} & (6)\end{matrix}$The approximation in (5) holds exactly when a response of a filter is anideal pulse function. The error of the approximation in (5) depends onthe shape of a response function and the width of in-band partition.

The measured k^(th) in-band integrated signal in (4) is a summation ofall average in-band signals that is given by

$\begin{matrix}{{\int_{\lambda \in {\{{\Delta\;\lambda_{l}}\}}}{{h_{k}(\lambda)}{s(\lambda)}\ {\mathbb{d}\lambda}}} \approx {\sum\limits_{l = 1}^{N}{{\overset{\_}{h}}_{k\; l}^{({i\; n})}\Delta\;\lambda_{l}{{\overset{\_}{s}}_{l}.}}}} & (7)\end{matrix}$The mean values s ₁ between λ_(min) ⁽¹⁾ and λ_(max) ⁽¹⁾ are the in-bandsignals we want to recover. The uncorrected (measured) signal is asuperposition of all in-band and band-gap signals. All coefficientfactors and parameters in equation (7) can be determined by the responsefunctions that are dependent on the characteristics of the filters for aparticular instrument.Band-Gap Partitions

To deal with the band-gap integral in equation (3), we consider that thek^(th) in-band spectral responses are much greater than the band-gapresponses as shown in FIG. 1, e.g.h _(k)(λ)(λ∈Δλ_(k))>>h _(k)(λ)(λ∉Δλ_(k)  (8)The measured signal in the band-gap integral can be interpolatedlinearly using the two nearest average bands [9] with nominal bandcenters λ₁ in Table 1 above.

$\begin{matrix}{{\hat{s}(\lambda)} = {{{\hat{s}}_{l}\frac{\left( {\lambda_{l + 1} - \lambda} \right)}{\left( {\lambda_{l + 1} - \lambda_{l}} \right)}} + {{\hat{s}}_{l + 1}\frac{\left( {\lambda - \lambda_{l}} \right)}{\left( {\lambda_{l + 1} - \lambda_{l}} \right)}\mspace{14mu}{\left( {\lambda_{l} \leq \lambda < \lambda_{l + 1}} \right).}}}} & (9)\end{matrix}$Assuming that the error between the k^(th) band measured and originalimages is given byε(λ)={circumflex over (s)}(λ)−s(λ),the band-gap integral in equation (3) becomes∫_(λ∉{Δλ1)}h_(k)(λ)s(λ)dλ=∫ _(λ∉{Δλ1) }h _(k)(λ)[ŝ(λ)+O(ε)]dλ≈∫_(λ∉{Δλ1) }h _(k)(λ)ŝ(λ)dλ.   (10)If we define λ_(max) ⁽⁰⁾=λ_(min) and λ_(min) ^((N+1))=λ_(max), andŝ(λ)=ŝ(λ_(min) ⁽¹⁾) (λ_(min)≦λ≦λ_(min) ⁽¹⁾) and ŝ(λ)=ŝ(λ_(max) ^((N)))(λ_(max) ^((N))≦λ≦λ_(max)), then the band-gap integral (10) using theinterpolation (9) can be formulated by

$\mspace{50mu}{{{\int_{\lambda \notin {\{{\Delta\;\lambda_{l}}\}}}{{h_{k}(\lambda)}{\hat{s}(\lambda)}\ {\mathbb{d}\lambda}}} \approx {\sum\limits_{l = 1}^{N}{b_{k\; l}{\hat{s}}_{l}}}},\mspace{79mu}{where}}$$b_{kl} = \left\{ {{{\begin{matrix}{{\left( {\lambda_{\min}^{(1)} - \lambda_{\min}} \right){\overset{\_}{h}}_{k,0}^{({out})}} + {\frac{\lambda_{\min}^{(2)} - \lambda_{\max}^{(1)}}{\lambda_{2} - \lambda_{1}}\left( {{\lambda_{2}{\overset{\_}{h}}_{k,1}^{({out})}} - {\overset{\_}{h\;\lambda}}_{k,1}^{({out})}} \right)}} & \left( {l = 1} \right) \\\begin{matrix}{{\frac{\lambda_{\min}^{(l)} - \lambda_{\max}^{({l - 1})}}{\lambda_{l} - \lambda_{l - 1}}\left( {{\overset{\_}{h\;\lambda}}_{k,{l - 1}}^{({out})} - {\lambda_{l - 1}{\overset{\_}{h}}_{k,{l - 1}}^{({out})}}} \right)} +} \\{\frac{\lambda_{\min}^{({l + 1})} - \lambda_{\max}^{(l)}}{\lambda_{l + 1} - \lambda_{l}}\left( {{\lambda_{l + 1}{\overset{\_}{h}}_{k\; l}^{({out})}} - {\overset{\_}{h\;\lambda}}_{k\; l}^{({out})}} \right)}\end{matrix} & {\left( {1 < l < N} \right),} \\\begin{matrix}{{\frac{\lambda_{\min}^{(N)} - \lambda_{\max}^{({N - 1})}}{\lambda_{N} - \lambda_{N - 1}}\left( {{\overset{\_}{h\;\lambda}}_{k,{N - 1}}^{({out})} - {\lambda_{N - 1}{\overset{\_}{h}}_{k,{N - 1}}^{({out})}}} \right)} +} \\{\left( {\lambda_{\max} - \lambda_{\max}^{(N)}} \right){\overset{\_}{h}}_{k\; N}^{({out})}}\end{matrix} & \left( {l = N} \right)\end{matrix}\mspace{20mu}{\overset{\_}{h\;\lambda}}_{k\; l}^{({out})}} = \frac{\int_{\lambda_{(\max)}^{(l)}}^{\lambda_{\min}^{({l + 1})}}{{h_{k}(\lambda)}\lambda\ {\mathbb{d}\lambda}}}{\lambda_{\min}^{({l + 1})} - \lambda_{\max}^{(l)}}},\mspace{20mu}{{{and}\mspace{20mu}{\overset{\_}{h}}_{k\; l}^{({out})}} = {\frac{\int_{\lambda_{(\max)}^{(l)}}^{\lambda_{\min}^{({l + 1})}}{{h_{k}(\lambda)}{\mathbb{d}\lambda}}}{\lambda_{\min}^{({l + 1})} - \lambda_{\max}^{(l)}}.}}} \right.$The linear approximation (9) between two nearest bands may cause a largeerror if real original signals m band-gap are far away from the linearapproximation curve. Fortunately, this error of signal in band-gap isnot our detected in-band signal and is convolved by a very low levelresponse function in the band-gap domain. The error in the hand-gapintegral can be ignored comparing with the in-hand integral instatistics since the properties of the responses between in-hand andband-gap domains in (9) hold for most cases.

Similarly, as in the above subsection, all coefficients b_(ki) andparameters λ_(min) ⁽¹⁾ and λ_(min) ⁽¹⁾ in equation (11) can h determinedand selected based on the response functions that are dependent on thecharacteristics of the filters for a particular instrument. We canadjust widths of the band-gap from zero to certain values for differentoptical instrument As shown in FIGS. 2A-2G, the widths of the band-gapsfrom M1 to M3 can be chosen as zero or smaller values, and larger widthsof the band gaps from M4 to M7 can improve the performance of the OOBcorrection for the VIIRS spectrometer.

Out-of-Band Correction Transform

Two terms of the in-band and band-gap integrals in equation (3) areformulated by equations (7) and (11). Then equation (3) can be rewrittenas

$\begin{matrix}{{\sum\limits_{l = 1}^{N}{{\overset{\_}{h}}_{k\; l}^{({i\; n})}\Delta\;\lambda_{l}{\overset{\_}{s}}_{l}}} = {{\hat{s}}_{k} - {\sum\limits_{l = 1}^{N}{b_{k\; l}{{\hat{s}}_{l}.}}}}} & (12)\end{matrix}$According to an embodiment of the instant invention, it is necessary tofind the, average in-band signals s _(i) twin the equation (12), inmatrix form, (12) is

${\begin{pmatrix}{\sum\limits_{l = 1}^{N}{{\overset{\_}{h}}_{1\; l}^{({i\; n})}\Delta\;\lambda_{l}{\overset{\_}{s}}_{l}}} \\{\sum\limits_{l = 1}^{N}{{\overset{\_}{h}}_{2\; l}^{({i\; n})}\Delta\;\lambda_{l}{\overset{\_}{s}}_{l}}} \\\ldots \\{\sum\limits_{l = 1}^{N}{{\overset{\_}{h}}_{N\; l}^{({i\; n})}\Delta\;\lambda_{l}{\overset{\_}{s}}_{l}}}\end{pmatrix} = \begin{pmatrix}{{\hat{s}}_{1} - {\sum\limits_{l = 1}^{N}{b_{1\; l}{\hat{s}}_{l}}}} \\{{\hat{s}}_{2} - {\sum\limits_{l = 1}^{N}{b_{2\; l}{\hat{s}}_{l}}}} \\\ldots \\{{\hat{s}}_{n} - {\sum\limits_{l = 1}^{N}{b_{N\; l}{\hat{s}}_{l}}}}\end{pmatrix}}\mspace{14mu}$ or${{A\overset{\_}{s}} = {\left( {I - B} \right)\hat{s}}},{where}$${\overset{\_}{s} = {{\begin{pmatrix}{\overset{\_}{s}}_{1} \\{\overset{\_}{s}}_{2} \\\ldots \\{\overset{\_}{s}}_{N}\end{pmatrix}\mspace{14mu}{and}\mspace{14mu}\hat{s}} = \begin{pmatrix}{\hat{s}}_{1} \\{\hat{s}}_{2} \\\ldots \\{\hat{s}}_{N}\end{pmatrix}}},$matrixes A=(h _(kj) ^((in))Δλ₁) and B=(b_(ki)) are N×N, and I is a N×Nidentity matrix. If the OBCT matrix T is defined byT=A ⁻(I−B),   (11)then the image by the OOB correction transform is given bys=Tŝ.   (14)Equation (14) is a linear transform between the uncorrected andcorrected multispectral image vectors.

The narrow-hand multispectral signals can be recovered from the measuredmultispectral signals, which contain OOB effects. The decompositionoperation can be simply performed by a product between a fixed OBCTmatrix and a measured multispectral image vector. All elements of theOBCT matrix T depend on the response functions of filters, in-bandwidths, and nominal band centers of the filters. Therefore, the OBCTmatrix can be fully determined by the characteristics of the filters fora particular multispectral radiometric instrument.

In the special case in which all filters are ideal, the normalizedresponse functions of the filters for the total wavelength range fromλ_(min) to λ_(max) are given by

${h_{k}(\lambda)} = {\frac{1}{{\Delta\lambda}_{l}}\left\{ \begin{matrix}1 & {\lambda_{\min}^{(l)} \leq \lambda \leq \lambda_{\max}^{(l)}} \\0 & {{otherwise}.}\end{matrix} \right.}$Using the spectral response function of the ideal filters, we found thatmatrix B=0, A=I, and the OBCT matrix T is an identity matrix. The inputand output signals are identical in this ideal system.VIIRS OBCT Matrix

Using equation (11), the recovered in-band signals can be calculated bythe OBCT matrix and the uncorrected multichannel image vector. In thissection we are concerned with a numerical computation of the OBCT matrixfor the VIIRS instrument.

The VIIRS instrument is, in many aspects, similar to the ModerateResolution Imaging Spectroradiometer (MODIS) instruments currently onboard the NASA Terra and Aqua Spacecrafts. Many VIIRS channels haveheritages to the MODIS instrument, but with minor differences in centerpositions and widths. Important differences between VIIRS and MODIS doexist. For example, VIIRS has five relatively broad imaging channels ata high spatial resolution of about 375 m.

All seven channel VIIRS filter transmittance functions shown in FIG. 1indicate that the filter bandwidths and positions are not uniform. Thetransmittance functions of the VIIRS filters in FIG. 1 are normalized toI at peak, points. All response functions defined in equation (1) and(3) must be normalized to an integral value of 1, as in Eq. (2), usingthe transmittance functions of the VIIRS filters in FIG. 1 before acomputation for the OBCT matrix. The normalized response functionsh_(k)(λ) are given by

${{h_{k}(\lambda)} = \frac{H_{k}(\lambda)}{\int_{\lambda_{\min}}^{\lambda_{\max}}{{H_{k}(\lambda)}\ {\mathbb{d}\lambda}}}},$where H_(k)(λ) are the transmittance functions of the VIIRS filters inFIG. 1 between wavelength range from λ_(min) to λ_(max).

The OBCT (7×7) matrix T fur the VIIRS instrument based on the wavelengthin-band and band-Rap partitions and the transmittance functions of thefilters in FIG. 1 is given by

$\begin{pmatrix}{\mspace{20mu} 1.0287} & {{- 1.70827} \times 10^{- 3}} & {{- 1.09021} \times 10^{- 4}} & {{- 5.55824} \times 10^{- 4}} & {{- 5.29429} \times 10^{- 3}} & {{- 2.75953} \times 10^{- 3}} & {{- 1.82264} \times 10^{- 2}} \\{{- 1.66515} \times 10^{- 3}} & {\mspace{20mu} 1.00991} & {{- 7.51183} \times 10^{- 4}} & {{- 1.41192} \times 10^{- 3}} & {{- 2.18476} \times 10^{- 3}} & {{- 9.52121} \times 10^{- 4}} & {{- 2.93506} \times 10^{- 3}} \\{{- 9.18167} \times 10^{- 4}} & {{- 6.31394} \times 10^{- 4}} & {\mspace{20mu} 1.01375} & {{- 1.41432} \times 10^{- 3}} & {{- 3.21069} \times 10^{- 3}} & {{- 1.78773} \times 10^{- 3}} & {{- 5.77032} \times 10^{- 3}} \\{{- 1.0098} \times 10^{- 3}} & {{- 4.86908} \times 10^{- 3}} & {{- 1.21268} \times 10^{- 2}} & {\mspace{20mu} 1.03442} & {{- 8.3853} \times 10^{- 3}} & {{- 4.17362} \times 10^{- 3}} & {{- 3.85384} \times 10^{- 3}} \\{{- 5.4833} \times 10^{- 4}} & {{- 1.07026} \times 10^{- 3}} & {{- 2.30808} \times 10^{- 3}} & {{- 4.43472} \times 10^{- 3}} & {\mspace{20mu} 1.01667} & {{- 3.45863} \times 10^{- 3}} & {{- 4.84276} \times 10^{- 3}} \\{{- 4.20704} \times 10^{- 4}} & {{- 4.19931} \times 10^{- 4}} & {{- 6.07297} \times 10^{- 4}} & {{- 9.42082} \times 10^{- 4}} & {{- 3.10139} \times 10^{- 3}} & {\mspace{14mu} 1.01041} & {{- 4.91217} \times 10^{- 3}} \\{{- 2.15101} \times 10^{- 4}} & {{- 1.42594} \times 10^{- 4}} & {{- 1.81492} \times 10^{- 4}} & {{- 1.91647} \times 10^{- 4}} & {{- 3.3216} \times 10^{- 4}} & {{- 1.56788} \times 10^{- 4}} & {\mspace{20mu} 1.00124}\end{pmatrix}\quad$All main diagonal elements in the OBCT matrix for the VIIRS instrumentare greater than but close to one. And all non-diagonal elements of theOBCT Matrix are negative because the uncorrected signal for a particularhand is a superposition of all in-band and OOB signals. The correctedsignal must be extracted from the superposition signals. The correctionamounts are dependent on the characteristics of the filters. The firstand fourth main diagonal elements with larger correction amounts(relative errors≈2.9% and 3.5%) in the OBCT matrix correspond to poorfilters such as band 1 and 4 as shown in FIGS. 2A-2G.

The summation of all elements in a row in the OBCT matrix is equal to 1,i.e.

${\sum\limits_{l = 1}^{N}T_{k\; l}} = 1.$Therefore, the correction coefficients in the OBCT matrix for each bandare also normalized.

To avoid overflow results for the matrix production between the OBCTmatrix and the spectral image vector, a data type of double precision isrecommended.

The method is based on the fact that other spectral channels measuresome of the light that contributes to OOB response in a particularchannel. This crosstalk between multispectral radiometers provides apossibility for decomposition. Using the filter transmittance functionsfor all multiband sensors, an OBCT matrix for recovering in-bandspectral radiance has been derived. The processing, of the OOBcorrection can be performed by a product between the OBCT matrix and amultispectral image vector.

The OBCT matrix for the Visible Infrared. Imager Radiometer Suite(“VIIRS”), which was successfully launched on Oct. 28, 2011, isnumerically computed and demonstrated. The VIIRS multispectral sensor isused as an example of the application of the method. Clearly, it can beapplied, to other multispectral sensors as well. In an embodiment of theinstant invention, the OBCT reduces the relative OOB errors in theuncorrected images by a factor of up to seventeen. An embodiment of theinvention can be applied to all multispectral remote sensing instrumentsfor OOB correction.

Optionally, VIIRS filter functions are obtained from measurements ofpre-launch laboratory platforms, high altitude aircraft platforms,and/or satellite platforms.

An embodiment of the invention comprises a computer program for imageprocessing, which computer program embodies the functions, filters, orsubsystems described herein. However, it should be apparent that therecould he many different ways of implementing the invention in computerprogramming, and the invention should not be construed as limited to anyone set of computer program instructions. Further, a skilled programmerwould be able to write such a computer program to implement an exemplaryembodiment based on the appended diagrams and associated description inthe application text. Therefore, disclosure of a particular set ofprogram code instructions is not considered necessary for an adequateunderstanding of how to make and use the invention. The inventivefunctionality of the claimed computer program will be explained in moredetail in the following description read in conjunction with the figuresillustrating the program flow.

One of ordinary skill in the art will recognize that the methods,systems, and control laws discussed above with respect to imageprocessing may be implemented in software as software modules orinstructions, in hardware (e.g., a standard field-programmable gatearray (“FPGA”) or a standard application-specific integrated circuit(“ASIC”), or in a combination of software and hardware. The methods,systems, and control laws described herein may be implemented on manydifferent types of processing devices by program code comprising programinstructions that are executable by one or more processors. The softwareprogram instructions may include source code, object code, machine code,or any other stored data that is operable to cause a processing systemto perform methods described herein.

The methods, systems, and control laws may be provided on many differenttypes of computer-readable media including computer storage mechanisms(e.g., CD-ROM, diskette, RAM, flash memory, computer's hard drive, etc.)that contain instructions fur use in execution by a processor to performthe methods operations and implement the systems described herein.

The computer components, software modules, functions and/or datastructures described, herein may be connected directly or indirectly toeach other in order to allow the flow of data needed for theiroperations, it is also ⁻toted that software instructions or a module canbe implemented for example as a subroutine unit or code, or as asoftware function unit of code, or as an object (as in anobject-oriented paradigm), or as an applet, or in a computer scriptlanguage, or as another type of computer code or firmware. The softwarecomponents and/or functionality may be located on a single device ordistributed across multiple devices depending upon the situation athand.

Systems and methods disclosed herein may use data signals conveyed usingnetworks (e.g., local area network, wide area network, internet, etc.),fiber optic medium, carrier waves, wireless networks, etc. forcommunication with one or more data processing devices. The data signalscan carry any or all of the data disclosed herein that is provided to orfrom a device.

This written description sets forth the best mode of the invention andprovides examples to describe the invention and to enable a person ofordinary skill in the art to make and use the in union. This writtendescription does not limit the in union to the precise terms set forth.Thus, while the invention has been described in detail with reference tothe examples set forth above, those of ordinary skill in the art mayeffect alterations, modifications and variations to the examples withoutdeparting from the scope of the invention.

These and other implementations are within the scope of the followingclaims.

What is claimed as new and desired to be protected by Letters Patent ofthe United States is:
 1. A method comprising: measuring a band-averagedspectral radiance with an N-channel multispectral sensor using at leastone optical filter, upon scanning, a plurality of original radiances,the measured total band-averaged spectral radiance,ŝ_(k)=ŝ_(k)(i,j)=∫_(λε{Δλ) ₁ }h _(k)(λ)s(λ)dλ+∫_(λε{Δλ) ₁ }h_(k)(λ)s(λ)dλ, where i and j are pixel indexes, comprising a measuredin-band-averaged spectral radiance and a measured band-gap-averagedspectral radiance, the measured total band-averaged spectral radiancecomprising a plurality of k wavelength sub-ranges, the plurality ofwavelength sub-ranges comprising a plurality of k in-band sub-ranges anda plurality a plurality of k band-gap sub-ranges, wherein a measured kthin-band band-averaged spectral radiance is represented as${{\int_{\lambda \in {\{{\Delta\;\lambda_{l}}\}}}{{h_{k}(\lambda)}{s(\lambda)}\ {\mathbb{d}\lambda}}} \approx {\sum\limits_{l = 1}^{N}{{\overset{\_}{h}}_{k\; l}^{({i\; n})}\Delta\;\lambda_{l}{\overset{\_}{s}}_{l}}}},$where N is a number of a plurality of bands, h _(kl) ^((in)) is anaverage of a plurality of filter response functions, Δλ₁ is a width ofpartitioned in-band sub-band, and s ₁ is a recovered Ith recoveredband-averaged spectral radiance that is an average of all measuredsignals within the sub-band Δλ₁, and free of an out-of-band effect,wherein a measured k-th band-gap band-average spectral radiance isrepresented as$\mspace{20mu}{{{\int_{\lambda \notin {\{{\Delta\;\lambda_{l}}\}}}{{h_{k}(\lambda)}{\hat{s}(\lambda)}\ {\mathbb{d}\lambda}}} \approx {\sum\limits_{l = 1}^{N}{b_{k\; l}{\hat{s}}_{l}}}},\mspace{79mu}{where}}$$b_{k\; l} = \left\{ {{{\begin{matrix}{{\left( {\lambda_{\min}^{(1)} - \lambda_{\min}} \right){\overset{\_}{h}}_{k,0}^{({out})}} + {\frac{\lambda_{\min}^{(2)} - \lambda_{\max}^{(1)}}{\lambda_{2} - \lambda_{1}}\left( {{\lambda_{2}{\overset{\_}{h}}_{k,1}^{({out})}} - {\overset{\_}{h\;\lambda}}_{k,1}^{({out})}} \right)}} & \left( {l = 1} \right) \\\begin{matrix}{{\frac{\lambda_{\min}^{(l)} - \lambda_{\max}^{({l - 1})}}{\lambda_{l} - \lambda_{l - 1}}\left( {{\overset{\_}{h\;\lambda}}_{k,{l - 1}}^{({out})} - {\lambda_{l - 1}{\overset{\_}{h}}_{k,{l - 1}}^{({out})}}} \right)} +} \\{\frac{\lambda_{\min}^{({l + 1})} - \lambda_{\max}^{(l)}}{\lambda_{l + 1} - \lambda_{l}}\left( {{\lambda_{l + 1}{\overset{\_}{h}}_{k\; l}^{({out})}} - {\overset{\_}{h\;\lambda}}_{k\; l}^{({out})}} \right)}\end{matrix} & {\left( {1 < l < N} \right),} \\\begin{matrix}{{\frac{\lambda_{\min}^{(N)} - \lambda_{\max}^{({N - 1})}}{\lambda_{N} - \lambda_{N - 1}}\left( {{\overset{\_}{h\;\lambda}}_{k,{N - 1}}^{({out})} - {\lambda_{N - 1}{\overset{\_}{h}}_{k,{N - 1}}^{({out})}}} \right)} +} \\{\left( {\lambda_{\max} - \lambda_{\max}^{(N)}} \right){\overset{\_}{h}}_{k\; N}^{({out})}}\end{matrix} & \left( {l = N} \right)\end{matrix}\mspace{20mu}{\overset{\_}{h\;\lambda}}_{k\; l}^{({out})}} = \frac{\int_{\lambda_{(\max)}^{(l)}}^{\lambda_{\min}^{({l + 1})}}{{h_{k}(\lambda)}\lambda\ {\mathbb{d}\lambda}}}{\lambda_{\min}^{({l + 1})} - \lambda_{\max}^{(l)}}},\mspace{20mu}{{{{and}\mspace{20mu}{\overset{\_}{h}}_{k\; l}^{({out})}} = \frac{\int_{\lambda_{(\max)}^{(l)}}^{\lambda_{\min}^{({l + 1})}}{{h_{k}(\lambda)}{\mathbb{d}\lambda}}}{\lambda_{\min}^{({l + 1})} - \lambda_{\max}^{(l)}}};}} \right.$generating from the measured band-average spectral radiance a measuredmultispectral radiance vector; matrix-multiplying the measuredmultispectral radiance vector and a out-of-band correction transformmatrix corresponding to the at least one optical filter to generate aband-average spectral radiances image vector representing a plurality ofrecovered band-averaged spectral radiance; outputting th plurality ofrecovered band-averaged spectral radiances, thereby generating aplurality of recovered radiances free of out-of-band effects; anddetermining a presence of a target based on the outputted plurality ofrecovered band-average spectral radiances.
 2. The method according toclaim 1, wherein each in-band sub-range of the plurality of k in-bandsub-ranges comprises an in-band width and each hand-gap sub-ranges ofthe plurality of k in-band sub-ranges comprises an band-gap width,wherein the at least one optical filter comprises at least one filtertransmittance function, wherein the plurality of sub-ranges comprises atleast one in-band partition parameter, and wherein the multispectraldecomposition transform matrix is a function of at least one of the atleast one filter transmittance function, the at least one partitionparameter, and a position of the at least one optical filter.
 3. Themethod according to claim 1, wherein the at least one optical filtercomprises a number of multi-bands, the number of multi-bands being equalto a number of the plurality of sub-ranges.
 4. The method according toclaim 1, wherein the recovered and measured band-averaged spectralradiance image vectors are represented as${\overset{\_}{s} = \begin{pmatrix}{\overset{\_}{s}}_{1} \\{\overset{\_}{s}}_{2} \\\ldots \\{\overset{\_}{s}}_{N}\end{pmatrix}},{{{and}\mspace{14mu}\hat{s}} = \begin{pmatrix}{\hat{s}}_{1} \\{\hat{s}}_{2} \\\ldots \\{\hat{s}}_{N}\end{pmatrix}}$ wherein each components s _(n) and ŝ_(n) of therecovered and measured band-averaged spectral radiance vectors aresingle hand images.
 5. The method according to claim 1, wherein therecovered band-averaged spectral radiance image vector is represented ass=Tŝ, wherein the out-of-baud correction transform matrix T is definedbyT=A ⁻¹(I−B), wherein matrix A=(h _(kl) ^((in))Δλ₁) and matrix B=(h_(kl))are N×N, wherein matrix I is a N×N identity matrix, wherein ŝ is themeasured band-averaged spectral radiance image vector.
 6. A methodaccording to claim 1, wherein the band-averaged spectral radiancecomprises a VIIRS band-averaged spectral radiance.
 7. The methodaccording to claim 1, wherein at least one optical filter resides on oneof an aircraft and a satellite.